The Use of Domination Number of a Random Proximity Catch Digraph for Testing Spatial Patterns of Segregation and Association

The Use of Domination Num b er of a Random Pro ximi t y Catc h Digraph for T esting Spatial P atterns of Segregation and Asso ciatio n ⋆ Elvan Ceyhan & Car ey E. Prie b e Johns Hopkins University, Baltimor e No v ember 14, 2018 Abstract Priebe et al. (2001) introduced the class cover c atch digraphs and computed the distribution of the domination num b er of such digraphs for one dimensional data. In higher d imensions these calculations are extremely difficult d ue to the geometry of th e proximit y regions; and only upp er-b ou n ds are av ailable. In th is article, we introdu ce a new type of data-rand om proximit y map and the asso ciated (di)graph in R d . W e fin d the asymptotic d istri bution of the domination number and u se it for testing spatial p oin t patterns of segregation and asso ciati on. Keywor ds: Random digraph; Domination num b er; Proximit y Map; Spatial Poin t Pattern; Segregation; Asso ciation; Dela unay T riangulation ⋆ This research was supp orted b y the Defense Adv anced Research Pro jects Agency as a dministered by the Air F orce Office of Scientific Research under contract DOD F49620 -99-1-021 3 and b y Office of Nav al Research Grant N000 14-95-1- 0777. Corresp onding author . E-mail addr ess: cep@jh u.edu (C.E. P r iebe) 1 1 In tro duction In a digr aph D = ( V , A ) with v ertex set V and arc (dire c ted edge) set A , a vertex v dominates itself and all vertices of the for m { u : v u ∈ A} . A dominating set , S D , for the digra ph D is a subset of V suc h that each vertex v ∈ V is dominated by a vertex in S D . A minimum dominating set , S ∗ D , is a dominating s et of minimum ca rdinalit y; and the domination n umb er , γ ( D ), is defined as γ ( D ) := | S ∗ D | , where | · | is the cardinality functional ([W es t, 20 01 ]). If a minimum dominating set is of size one, we call it a dominating p oint . Let (Ω , M ) be a measur able s pace a nd consider a function N : Ω × 2 Ω → 2 Ω , where 2 Ω represents the power set of Ω. Then given Y ⊆ Ω, the pr oximity map N Y ( · ) = N ( · , Y ) : Ω → 2 Ω asso ciates with each po in t x ∈ Ω a pr oximity re gion N Y ( x ) ⊂ Ω. The regio n N Y ( x ) dep ends on the distance b et ween x and Y . F or B ⊆ Ω, the Γ 1 -r e gion , Γ 1 ( · ) = Γ 1 ( · , N Y ) : Ω → 2 Ω asso ciates the region Γ 1 ( B ) := { z ∈ Ω : B ⊆ N Y ( z ) } with each set B ⊆ Ω. F o r x ∈ Ω, we denote Γ 1 ( { x } ) as Γ 1 ( x ). If X n = { X 1 , X 2 , · · · , X n } is a set of Ω-v alued random v ariables , then the N Y ( X i ) (and Γ 1 ( X i )), i = 1 , · · · , n are ra ndom sets. If the X i are indep enden t a nd ident ically distributed, then so are the random sets N Y ( X i ) (and Γ 1 ( X i )). F urther more, Γ 1 ( X n ) is a r andom set. Notice that Γ 1 ( X n ) = ∩ n j =1 Γ 1 ( X j ), sinc e x ∈ Γ 1 ( X n ) iff X n ⊆ N Y ( x ) iff X j ∈ N Y ( x ) for all j = 1 , . . . , n iff x ∈ Γ 1 ( X j ) for all j = 1 , . . . , n iff x ∈ ∩ n j =1 Γ 1 ( X j ). Consider the data -random proximity catch digra ph D with vertex set V = X n and ar c se t A defined b y ( X i , X j ) ∈ A ⇐ ⇒ X j ∈ N Y ( X i ). The random digraph D dep ends on the (joint) distribution o f the X i and on the map N Y (see Pr iebe et al. (2001) and Prieb e et a l. (2 003)). The adjective pr oximity — for the ca tc h digraph D and for the map N Y — comes fro m thinking of the region N Y ( x ) as repre s en ting those p oin ts in Ω “clo s e” to x (see, e.g., T oussa int (198 0 ) and Jar omczyk and T ouss ain t (19 92)). F or X 1 , · · · , X n iid ∼ F the domination num b er of the a ssocia ted da ta-random proximit y catch digraph D , deno ted γ ( X n ; F, N Y ), is the minimum n umber of p oin ts that dominate all p oints in X n . Note that, γ ( X n ; F, N Y ) = 1 iff X n ∩ Γ 1 ( X n ) 6 = ∅ . The ra ndom v aria ble γ n := γ ( X n ; F, N Y ) depends on X n explicitly , a nd on F and N Y implicitly . In general, the exp ectation E [ γ n ], dep ends o n n , F , and N Y ; 1 ≤ E [ γ n ] ≤ n ; and the v ar iance of γ n satisfies, 0 ≤ V ar [ γ n ] ≤ n 2 / 4. W e can also define the reg ions asso ciated with γ n = k for k ≤ n . F o r insta nce, the Γ 2 -region for proximity map N Y ( · ) and set B ⊂ Ω is Γ 2 ( B ) = { ( x, y ) ∈ [Ω \ Γ 1 ( B )] 2 : B ⊆ N Y ( x ) ∪ N Y ( y ) } . In general, Γ k ( B ) = { ( x 1 , x 2 , . . . , x k ) ∈ Ω k : B ⊆ ∪ k j =1 N Y ( x j ) and all p ossible m -p ermut ations ( u 1 , u 2 , . . . , u m ) of ( x 1 , x 2 , . . . , x k ) satisfy ( u 1 , u 2 , . . . , u m ) 6∈ Γ m ( B ) for each m = 1 , 2 , . . . , k − 1 } . 2 A Class of P ro ximit y Maps and the Corresp onding Γ 1 -Regions Let Ω = R 2 and let Y = { y 1 , y 2 , y 3 } ⊂ R 2 be thr ee non-co llinear po in ts. Denote by T ( Y ) the triangle —including the interior— formed by these three points. The most s traigh tforward extension of the data random proximity catch digraph introduced by Prieb e et al. (2001) is the spherical pr o ximit y map N S ( x ) = B ( x, r ( x )) which is the ball centered at x with radius r ( x ) = min y ∈Y d ( x, y ) or the a rc-slice proximity map N AS ( x ) = B ( x, r ( x ) ) ∩ T ( Y ). How ever, b oth case s suffer from the intractabilit y o f the Γ 1 -region a nd hence the intractability of the finite and asymptotic distribution of γ n . W e pr opose a new clas s of pr o x imit y regio ns which do es not suffer fro m this drawback. F or r ∈ [1 , ∞ ] define N r Y to b e the r-factor pr oximity map and Γ r 1 to b e the corresp onding Γ 1 -region as follows; see also Fig ur es 1 a nd 2. Let “ v er tex regions ” R ( y 1 ), R ( y 2 ), R ( y 3 ) partition T ( Y ) using seg - men ts from the cen ter of mass of T ( Y ) to the edge midp oints. F or x ∈ T ( Y ) \ Y , let v ( x ) ∈ Y b e the 2 vertex whos e reg ion contains x ; x ∈ R ( v ( x )). If x falls on the b oundary of tw o vertex regions, we assign v ( x ) ar bitrarily . Let e ( x ) be the edge of T ( Y ) opp osite v ( x ). Le t ℓ ( v ( x ) , x ) b e the line para lle l to e ( x ) through x . Let d ( v ( x ) , ℓ ( v ( x ) , x )) b e the Euclidean (p e r pendicular) distance from v ( x ) to ℓ ( v ( x ) , x ). F or r ∈ [1 , ∞ ) let ℓ r ( v ( x ) , x ) be the line pa rallel to e ( x ) such that d ( v ( x ) , ℓ r ( v ( x ) , x )) = rd ( v ( x ) , ℓ ( v ( x ) , x )) a nd d ( ℓ ( v ( x ) , x ) , ℓ r ( v ( x ) , x )) < d ( v ( x ) , ℓ r ( v ( x ) , x )). Let T r ( x ) be the triangle s imila r to and with the same o ri- ent ation as T ( Y ) having v ( x ) a s a vertex and ℓ r ( v ( x ) , x ) as the opp osite edge. Then the r-factor proximit y region N r Y ( x ) is defined to b e T r ( x ) ∩ T ( Y ). T o define the Γ 1 -region, let ξ j ( x ) b e the line suc h that ξ j ( x ) ∩ T ( Y ) 6 = ∅ and r d ( y j , ξ j ( x )) = d ( y j , ℓ ( y j , x )) for j = 1 , 2 , 3 . Then Γ r 1 ( x ) = ∪ 3 j =1  Γ r 1 ( x ) ∩ R ( y j )  where Γ r 1 ( x ) ∩ R ( y j ) = { z ∈ R ( y j ) : d ( y j , ℓ ( y j , z )) ≥ d ( y j , ξ j ( x ) } , for j = 1 , 2 , 3 . No tice that r ≥ 1 implies x ∈ N r Y ( x ) and x ∈ Γ r 1 ( x ). F ur thermore, lim r →∞ N r Y ( x ) = T ( Y ) and lim r →∞ Γ r 1 ( x ) = T ( Y ) for all x ∈ T ( Y ) \ Y , and s o we define N ∞ Y ( x ) = T ( Y ) and Γ ∞ 1 ( x ) = T ( Y ) for all such x . F or x ∈ Y , we define N r Y ( x ) = { x } for all r ∈ [1 , ∞ ]. Notice that X i iid ∼ F , with the additional as sumption that the non- de g enerate tw o -dimensional probability density function f exists with supp ort( f ) ⊆ T ( Y ), implies tha t the s pecial case in the construction of N r Y — X falls o n the bo undary of tw o vertex r egions — o ccurs with proba bilit y zer o. Note that for such an F , N r Y ( x ) is a triangle a.s. and Γ r 1 ( x ) is a s ta r-shape d p olygon (not neces sarily conv ex). y 1 = v ( x ) x M C ℓ ( v ( x ) , x ) ℓ 2 ( v ( x ) , x ) y 3 e ( x ) y 2 d ( v ( x ) , ℓ 2 ( v ( x ) , x )) = 2 d ( v ( x ) , ℓ ( v ( x ) , x )) d ( v ( x ) , ℓ ( v ( x ) , x )) Figure 1: Construction o f r -factor proximit y regio n, N 2 Y ( x ) (shaded region). Let X e := a rgmin X ∈X n d ( X, e ) b e the (closest) edge extremum for edge e . Then Γ r 1 ( X n ) = ∩ 3 j =1 Γ r 1 ( X e j ), where e j is the edge opp osite vertex y j , for j = 1 , 2 , 3 . So Γ r 1 ( X n ) ∩ R ( y j ) = { z ∈ R ( y j ) : d ( y j , ℓ ( y j , z ) ≥ d ( y j , ξ j ( X e j )) } , for j = 1 , 2 , 3. Let the domination num b er b e γ n ( r ) := γ n ( X n ; F, N r Y ) and X [ j ] := arg min X ∈X n ∩ R ( y j ) d ( X, e j ). Then γ n ( r ) ≤ 3 with probability 1, since X n ∩ R ( y j ) ⊂ N r Y ( X [ j ] ) for each j = 1 , 2 , 3. Thus 1 ≤ E [ γ n ( r )] ≤ 3 and 0 ≤ V ar [ γ n ( r )] ≤ 9 / 4 . 3 y 1 y 3 ℓ ( y 1 , x ) x ξ 3 (2 , x ) ξ 2 (2 , x ) d ( y 1 , ℓ ( y 1 , x )) = r d ( y 1 , ξ 1 (2 , x )) d ( y 1 , ξ 1 (2 , x )) ξ 1 (2 , x ) y 2 M C Figure 2: Construction o f the Γ 1 -region, Γ 2 1 ( x ) (shaded r e g ion). 3 Null Distribution of Domination Num b er The nu ll hypothesis for spatia l patterns hav e been a co n traversial topic in eco logy from the ear ly days. [Gotelli and Graves, 1996] hav e collected a voluminous literature to present a compr ehensiv e analysis o f the use and misuse of null mo dels in eco logy comm unit y . They also define and attempt to clarify the null mo del concept as “a pattern-ge nerating mo del that is based on ra ndomization o f ecolo gical data or random sampling from a known o r imagined distr ibution. . . . The ra ndo mization is designed to pro duce a pa ttern that w ould be exp ected in the absence of a par ticular ecolo gical mechanism.” In other words, the h yp othesized null mo dels can b e viewed as “ tho ugh t exp eriment s,” which is conv ent ially us ed in the physical sciences , and these mo dels pr o vide a statistical baseline fo r the analysis o f the pa tterns. F or statistical testing, the nu ll hypothesis we cons ider is a type o f c omplete sp atial r andomness ; that is, H 0 : X i iid ∼ U ( T ( Y )) where U ( T ( Y )) is the unifor m dis tribution on T ( Y ). If it is desired to hav e the sample size b e a random v ariable, we may consider a spa tial Poisson p oint pro cess on T ( Y ) as our null hypothesis. W e first present a “g eometry inv ar iance” r esult which allows us to a ssume T ( Y ) is the s tandard eq uila teral triangle, T  (0 , 0) , (1 , 0) ,  1 / 2 , √ 3 / 2  , thereby simplifying our subsequent analysis. Theorem 1 : Let Y = { y 1 , y 2 , y 3 } ⊂ R 2 be three non-collinear p oin ts. F or i = 1 , · · · , n , let X i iid ∼ F = U ( T ( Y )), the unifor m distr ibutio n o n the triangle T ( Y ). Then for any r ∈ [1 , ∞ ] the distribution of γ ( X n ; U ( T ( Y )) , N r Y ) is indep enden t o f Y , and hence the geo metry of T ( Y ). Pro of: A comp osition of tra nslation, r otation, re fle c tio ns, a nd sca ling will ta k e a n y given triang le T o = T ( { y 1 , y 2 , y 3 } ) to the “ basic” tr iangle T b = T ( { (0 , 0) , (1 , 0) , ( c 1 , c 2 ) } ) with 0 < c 1 ≤ 1 / 2, c 2 > 0 a nd (1 − c 1 ) 2 + c 2 2 ≤ 1, preserving uniformity . The transfor ma tion φ e : R 2 → R 2 given by φ e ( u, v ) =  u + 1 − 2 c 1 √ 3 v , √ 3 2 c 2 v  takes T b to the e quilateral tr iangle T e = T ( { (0 , 0) , (1 , 0) , (1 / 2 , √ 3 / 2) } ). Inv es tig ation of the Jaco bian shows that φ e also pre serv es uniformity . F urther mo re, the comp osition of φ e with the r igid motion transforma tio ns maps the b oundary of the orig inal tr iangle, T o , to the b oundary of the equilater al tr iangle, T e , the median lines of T o to the median lines of T e , a nd lines para llel to the edges of T o to lines par allel to the edg es of T e . 4 k  n 10 20 30 40 50 60 70 80 90 100 200 300 1 151 82 61 6 7 50 24 29 21 15 27 10 7 2 602 636 688 670 693 714 739 7 08 723 718 715 730 3 247 282 251 263 257 262 232 2 71 262 255 275 263 T able 1: The num b e r o f γ n (3 / 2) = k out of N = 100 0 replicates. Since the distribution of γ ( X n ; U ( T ( Y )) , N r Y ) inv o lves only pro babilit y co n tent o f unions and intersections of r egions b ounded by precisely such lines, and the pro babilit y conten t of such re g ions is pr eserved since uniformity is pr eserv ed, the desired result follows.  Based on Theorem 1 and our unifor m null hypothesis, w e may assume that T ( Y ) is a standard equilater a l triangle with Y = { (0 , 0 ) , (1 , 0) , (1 / 2 , √ 3 / 2) } henceforth. F or our r -factor proximity map and uniform n ull h yp othesis, the asymptotic n ull distribution of γ n ( r ) := γ ( X n ; U ( T ( Y )) , N r Y ) ca n be der iv ed as a function o f r . W e denote by ς r Y := { z ∈ T ( Y ) : N r Y ( z ) = T ( Y ) } the sup erset r e gion ass ociated with N r Y in T ( Y ). Notice tha t ς r Y ⊆ Γ r 1 ( X n ) for all r and X n ∩ ς r Y 6 = ∅ implies that γ n ( r ) = 1 . Prop osition 1: The exp ected area of the the Γ 1 -region, E [ A (Γ r 1 ( X n ))], c o n verges to the area of the sup e rset reg ion, A ( ς r Y ), as n → ∞ . In pa rticular, E [ A (Γ 3 / 2 1 ( X n ))], go es to zero at r ate O ( n − 2 ) as n → ∞ . Pro of: See App endix.  As a co rollary to the ab o ve prop osition, we hav e that E [ A (Γ r 1 ( X n ))] → A ( ς r Y ) = 0 for r ∈ [1 , 3 / 2] as n → ∞ . Additionally , E [ A (Γ r 1 ( X n ))] → A ( ς r Y ) = (1 − 3 / (2 r )) 2 √ 3 for r ∈ (3 / 2 , 2], and E [ A (Γ r 1 ( X n ))] → A ( ς r Y ) = √ 3 / 4 (1 − 3 / r 2 ) for r ∈ (2 , ∞ ], as n → ∞ . Theorem 2: The domination num b er γ n ( r ) = γ ( X n ; U ( T ( Y )) , N r Y ) is degener ate in the limit for r ∈ [1 , ∞ ] \ { 3 / 2 } as n → ∞ . Pro of: F or r ∈ [1 , 3 / 2), ς r Y = ∅ and T ( Y ) \ N r Y ( X [ j ] ) has pos itiv e area fo r all j = 1 , 2 , 3. F ur thermore, T ( Y ) \ ( N r Y ( X [ j ] ) ∪ N r Y ( X [ k ] )) ha s p ositive area fo r all pa irs { j, k } ⊂ { 1 , 2 , 3 } . Reca ll that γ n ( r ) ≤ 3 with probability 1 for all n a nd r . Hence γ n ( r ) → 3 in probability a s n → ∞ . F or r ∈ (3 / 2 , ∞ ], ς r Y has p ositiv e area , so γ n ( r ) → 1 in probability as n → ∞ .  Theorem 3 : F or r = 3 / 2 , lim n →∞ γ n ( r ) > 1 a.s. In pa r ticular lim n →∞ γ n (3 / 2) = ( 2 wp ≈ . 7 413 , 3 wp ≈ . 2 487 . Thu s E [ γ n (3 / 2)] → µ ≈ 2 . 2587 a s n → ∞ , and V ar [ γ n (3 / 2)] → σ 2 ≈ . 191 8 as n → ∞ . Pro of: See App endix.  The finite sa mple distribution of γ n (3 / 2), and hence the finite s ample mean and v aria nce, can b e obtained by num erical metho ds. W e estimate the distr ibution of γ n (3 / 2) fo r v ar ious fixed n empirically . In T able 1, we present empirica l estima tes for n = 10 , 2 0 , . . . , 10 0 , 20 0 , 30 0 with 1000 Monte Carlo r e plicates. See also Figure 3. Theorem 4 Let γ n ( r ) = γ ( X n ; U ( T ( Y )) , N r Y ). Then r 1 < r 2 implies γ n ( r 2 ) < S T γ n ( r 1 ). Pro of: Supp ose r 1 < r 2 . Then P ( γ n ( r 2 ) = 1) > P ( γ n ( r 1 ) = 1) a nd P ( γ n ( r 2 ) = 2) > P ( γ n ( r 1 ) = 2) and P ( γ n ( r 2 ) = 3 ) < P ( γ n ( r 1 ) = 3). Hence the desired res ult follows.  5 0 50 100 150 200 250 300 0.0 0.2 0.4 0.6 0.8 1.0 n empirical value of P( γ n =k) k=1 k=2 k=3 Figure 3: Plotted a re the empirical estimates of P ( γ n (3 / 2) = k ) versus v arious n v alues. 4 The Null Distribution of the Mean Domination Num b er in the Multiple T riangle Case Suppo se Y is a finite co llection of po in ts in R 2 with |Y | ≥ 3 . Consider the Delaunay triangulation (assumed to exist) of Y , where T j denotes the j th Delaunay triangle , J denotes the n umber of tr iangles, a nd C H ( Y ) denotes the co nvex h ull of Y (Ok ab e et al. (2000)). W e wis h to inv estigate H 0 : X i iid ∼ U ( C H ( Y )) against segreg ation and asso ciation alternatives (see Section 5). Figure 4 presents a realizatio n o f 100 0 o bs erv atio ns indep enden t a nd iden tically dis tributed according to U ( C H ( Y )) for |Y | = 10 and J = 13 . 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 Figure 4: A re a lization of H 0 for |Y | = 10, J = 13 , a nd n = 100 0 . The digra ph D is constructed using N r Y j ( · ) as des c ribed a bov e, where for X i ∈ T j the three p oints in Y defining the Dela una y tria ngle T j are used as Y j . Let γ n j ( r ) b e the domination num b er of the comp onen t of the dig raph in T j , w he r e n j = |X n ∩ T j | . 6 Theorem 5: (Asymptotic Normality) Supp ose n j ≫ 1 and J is sufficien tly lar ge. Then the null distribution of the mean dominatio n num b er G J := 1 J P J j =1 γ n j (3 / 2) is given by G J approx ∼ N ( µ, σ 2 /J ) where µ a nd σ 2 are given in Theo rem 3 a bov e. Pro of: F or fixed J sufficien tly large and each n j sufficiently lar ge, γ n j (3 / 2) are approximately indep en- dent identically distr ibuted as in Theo rem 3.  Figure 5 indicates that, for J = 13 with the realization of Y given in Figure 4 and n = 100 the nor mal approximation is not appropria te, even though the distribution lo oks symmetric, since no t a ll n j are suf- ficient ly larg e , but for n = 10 00 the histog ram and the corre s ponding normal curve are similar indicating that this sa mple size is lar ge enoug h to allow the use of the a symptotic no rmal approximation, since all n j are sufficient ly larg e . How ever, larg er J v alues require larg e r s ample sizes in or der to obtain approximate normality . 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 P S f r a g r e p la c e m e n t s Density 1.8 2.0 2.2 2.4 2.6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 P S f r a g r e p la c e m e n t s Density Figure 5: Depicted ar e G J approx ∼ N ( µ ≈ 2 . 258 7 , σ 2 /J ≈ . 19 17 /J ) for J = 13 and n = 100 (left) n = 10 00 (right). Histograms are based on 1000 Monte Ca rlo replicates and the curves are the a ssociated approximating normal curves. F or finite n , le t G J ( r ) be the mean do mination n umber a ssocia ted with the digr a ph based on N r Y . Then as a cor ollary to Theo rem 4 it follows that for r 1 < r 2 , we hav e G J ( r 2 ) < S T G J ( r 1 ). 5 Alternativ es: S egre gation and Asso ciation In a tw o class setting, the phenomenon known as se gr e gation o ccurs when members of one class ha ve a tendency to r epel members o f the other class. F or ins tance, it may b e the case that one t yp e of plant do es not grow w ell in the vicinity of another type of plant, and vice versa. This implies , in our notation, that X i are unlikely to b e lo cated near an y elemen ts of Y . Alternatively , asso ciation o ccurs when members of one class hav e a tendency to attract members of the other clas s , as in sy m biotic sp ecies, so that the X i will tend to cluster a round the elements of Y , for example. See, fo r instance, [Dixon, 1994], [Co omes et al., 1 999 ]. W e define tw o simple class e s of alternatives, H S ε and H A ε with ε ∈ (0 , √ 3 / 3), for segr egation and asso- ciation, res pectively . Let Y e = { (0 , 0) , (0 , 1) , (1 / 2 , √ 3 / 2) } and T e = T ( Y e ). F or y ∈ Y e , let e ( y ) denote the edge o f T e opp osite vertex y , and for x ∈ T e let ℓ y ( x ) denote the line par allel to e ( y ) through x . Then define T ( y , ε ) = { x ∈ T e : d ( y , ℓ y ( x )) ≤ ε } . Let H S ε be the model under which X i iid ∼ U ( T e \ ∪ y ∈Y T ( y , ε )) and H A ε be the mo del under which X i iid ∼ U ( ∪ y ∈Y T ( y , √ 3 / 3 − ε )). Thu s the segr egation mo del excludes the p ossibility of any X i o ccurring near a y j , a nd the as sociatio n mo del req uires that all X i o ccur near y j ’s. The √ 3 / 3 − ε in the definition of the ass ociation alternative is so that ε = 0 yields H 0 under b oth cla sses o f a lternativ es. 7 Remark: These definitions of the alternatives are g iv en for the standard equila teral triangle. The geometry inv ar iance result of Theorem 1 still holds under the alternatives, in the following sense. If, in a n arbitrar y triangle, a small p ercen tage δ · 100% where δ ∈ (0 , 4 / 9) of the ar ea is carved awa y as forbidden from each v ertex using line seg men ts parallel to the opp osite edg e, then under the transfor mation to the standard equilateral triangle this will result in the alternative H S √ 3 δ/ 4 . This arg ument is for seg regation; a similar constr uction is av a ilable for asso ciation. Theorem 6: (Stochastic Ordering) Let γ n,ε ( r ) b e the domination num b er under the seg regation alter- native with ε > 0 . Then with ε j ∈ (0 , √ 3 / 3), j = 1 , 2, ε 1 > ε 2 implies that γ n,ε 1 (3 / 2) < S T γ n,ε 2 (3 / 2). Pro of: Note that P ( γ n,ε 1 (3 / 2) = 1 ) > P ( γ n,ε 2 (3 / 2) = 1 ) and P ( γ n,ε 1 (3 / 2) = 2 ) > P ( γ n,ε 2 (3 / 2) = 2 ), hence the de s ired res ult follows.  Note that for Theore m 6 to hold in the limiting ca se, ε 1 ∈ (0 , √ 3 / 4] and ε 2 ∈ ( √ 3 / 4 , √ 3 / 3) should hold. F or ε ∈ (0 , √ 3 / 4], γ n,ε (3 / 2) → 2 in pr obabilit y a s n → ∞ , and for ε ∈ ( √ 3 / 4 , √ 3 / 3), γ n,ε (3 / 2) → 1 in probability as n → ∞ . Similarly , the sto c hastic order ing result of The o rem 6 holds for asso ciation for all ε and n < ∞ , with the inequalities b eing reversed. Notice that under seg regation with ε ∈ (0 , √ 3 / 4), γ n,ε ( r ) is degenerate in the limit e xcept for r = (3 − √ 3 ε ) / 2. With ε ∈ ( √ 3 / 4 , √ 3 / 3), γ n,ε ( r ) is degenera te in the limit except for r = √ 3 /ε − 2. Under asso ciation with ε ∈ (0 , √ 3 / 4), γ n,ε ( r ) is degener a te in the limit except for r = 3 2 (1 − √ 3 ε ) . The mean domination n umber of the proximit y catch dig raph, G J := 1 J P J j =1 γ n j (3 / 2), is a test statistic for the seg regation/ass o ciation a lternativ e; r ejecting for extreme v alues of G J is a ppropriate, s ince under segrega tion we exp ect G J to b e small, while under as sociatio n we exp ect G J to b e large. Using the equiv alent test statistic S = √ J ( G J − µ ) /σ, (1) the asymptotic cr itical v alue for the o ne-sided level α test ag ainst segreg a tion is given by z 1 − α = Φ − 1 ( α ) (2) where Φ( · ) is the sta ndard normal distribution function. The test r ejects for S < z 1 − α . Aga ins t asso ciation, the test r ejects for S > z α . Depicted in Figur e 6 ar e the segr egation with δ = 1 / 16 a nd ass ociation with δ = 1 / 4 r ealizations for |Y | = 10 and J = 1 3, and n = 10 00. The asso ciated mean domination n umbers ar e 2 . 308 , 1 . 923, and 3 . 000 , for the null realiza tion in Figure 4 a nd the se g regation a nd a ssocia tio n alterna tives in Figure 6, resp ectively , yielding p-v alues . 660 , . 003 a nd 0 . 000 . W e also pre s en t a Monte Car lo p o wer inv estiga tion in Section 6 for these cas e s. Theorem 7 : (Consistency) Let J ∗ ( α, ε ) :=   σ · z α µ − G J  2  where ⌈ ·⌉ is the ceiling function and ε -dep endence is throug h G J under a g iv en a lternativ e. Then the test aga inst H S ε which rejects for S < z 1 − α is cons is ten t for all ε ∈ (0 , √ 3 / 3) and J ≥ J ∗ (1 − α, ε ), and the test a gainst H A ε which rejects for S > z α is consistent for all ε ∈ (0 , √ 3 / 3) and J ≥ J ∗ ( α, ε ). Pro of: Let ε > 0. Under H S ε , γ n,ε (3 / 2) is degene r ate in the limit as n → ∞ , which implies G J is a constant a.s. In particula r , for ε ∈ (0 , √ 3 / 4], G J = 2 and for ε ∈ ( √ 3 / 4 , √ 3 / 3), G J = 1 a.s. as n → ∞ . Then the test statistic S = √ J ( G J − µ ) / σ is a c onstan t a.s. and J ≥ J ∗ (1 − α, ε ) implies that S < z 1 − α a.s. Hence consis tency follows for segre g ation. Under H A ε , as n → ∞ , G J = 3 for all ε ∈ (0 , √ 3 / 3), a.s. Then J ≥ J ∗ ( α, ε ) implies that S > z α a.s., hence consis tency follows for as sociatio n.  8 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 Figure 6: A re a lization of se g regation (left) a nd asso ciation (r igh t) for |Y | = 10, J = 13 , a nd n = 100 0. 6 Mon te Carl o P o w er Analysis In Figure 7, w e observe empirically that even under mild seg regation we obtain co nsiderable sepa ration betw een the kernel density estimates under null and segr e gation ca ses for mo derate J and n v a lues sugges ting high power a t α = . 0 5. A similar r esult is obs erv ed for as s ociation. With J = 1 3 a nd n = 1000 , under H 0 , the estimated significa nce lev el is b α = . 0 9 relative to segr egation, and b α = . 07 rela tiv e to asso ciation. Under H S √ 3 / 8 , the empirical power (using the a symptotic cr itica l v alue) is b β = . 97, and under H A √ 3 / 21 , b β = 1 . 00. With J = 30 and n = 5000, under H 0 , the estimated sig nificance level is b α = . 06 relative to seg regation, and b α = . 04 relative to a ssocia tion. The empirical p o wer is b β = 1 . 00 for b oth alternatives. W e also estimate the empiric a l p o wer b y using the empirica l critical v a lues. With J = 13 and n = 1 000, under H S √ 3 / 8 , the empirical p ow er is b β mc = . 7 2 at empirical level b α mc = . 0 33 and under H A √ 3 / 21 the empirical power is b β mc = 1 . 00 at empirica l level b α mc = . 03 . With J = 30 and n = 5 000, under H S √ 3 / 8 , the empirical power is b β mc = 1 . 0 0 at empirical level b α mc = . 03 4 and under H A √ 3 / 21 the empirical p o wer is b β mc = 1 . 0 0 at empirical level b α mc = . 0 4. 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 0 1 2 3 4 5 P S f r a g r e p la c e m e n t s kernel density es timate relative density 1.8 2.0 2.2 2.4 2.6 0 2 4 6 8 10 P S f r a g r e p la c e m e n t s kernel density es timate relative density Figure 7: Two Mo n te Carlo ex periments agains t the seg regation alternatives H S √ 3 / 8 with δ = 1 / 16. Depicted are kernel density estimates of G J for J = 1 3 and n = 100 0 with 1 0 00 replicates (left) and J = 30 and n = 5000 with 100 0 replicates (rig ht) under the null (solid) and alter nativ e (dashed). 9 7 Extension to Hig her Dimensions The e xtension to R d for d > 2 is straightforw ard. Let Y = { y 1 , y 2 , · · · , y d +1 } b e d + 1 no n-coplanar p oin ts. Denote the simplex for med by these d + 1 p oint s as S ( Y ). (A simplex is the simplest po lytope in R d having d + 1 v ertices, d ( d + 1) / 2 edges , and d + 1 faces of dimension ( d − 1).) F or r ∈ [1 , ∞ ], define the r -factor proximit y map as follows. Given a po in t x in S ( Y ), let y := arg min y ∈Y volume ( Q y ( x )) where Q y ( x ) is the p olytope with vertices b eing the d ( d + 1) / 2 midp oint s of the edg es, the vertex y and x . That is, the vertex r egion for vertex v is the p olytope with vertices g iven by v a nd the midp oin ts of the edges. Let v ( x ) be the vertex in whose r egion x falls . If x falls on the bo undary o f tw o vertex regions, we assign v ( x ) arbitra rily . Let ϕ ( x ) b e the face opp osite to vertex v ( x ), and η ( v ( x ) , x ) b e the hyper pla ne par allel to ϕ ( x ) which contains x . Let d ( v ( x ) , η ( v ( x ) , x )) b e the (p erpendicular ) Euclidea n distance fro m v ( x ) to η ( v ( x ) , x ). F or r ∈ [1 , ∞ ), let η r ( v ( x ) , x ) be the hyperplane para lle l to ϕ ( x ) such that d ( v ( x ) , η r ( v ( x ) , x )) = r d ( v ( x ) , η ( v ( x ) , x )) a nd d ( η ( v ( x ) , x ) , η r ( v ( x ) , x )) < d ( v ( x ) , η r ( v ( x ) , x )). Le t S r ( x ) b e the p olytop e similar to and with the sa me o rien tation a s S ha ving v ( x ) as a vertex and η r ( v ( x ) , x ) as the oppo site face. Then the r - factor proximit y region N r Y ( x ) := S r ( x ) ∩ S ( Y ). Also, let ζ j ( x ) b e the h yp erplane such that ζ j ( x ) ∩ S ( Y ) 6 = ∅ and r d ( y j , ζ j ( x )) = d ( y j , η ( y j , x )) for j = 1 , 2 , . . . , d + 1. Then Γ r 1 ( x ) = ∪ d +1 j =1 (Γ r 1 ( x ) ∩ R ( y j )) where Γ r 1 ( x ) ∩ R ( y j ) = { z ∈ R ( y j ) : d ( y j , η ( y j , z )) ≥ d ( y j , ζ j ( x ) } , for j = 1 , 2 , 3 . Theorem 1 g eneralizes, so that any simplex S in R d can b e tr a nsformed in to a regular poly tope (with egdes b eing equal in length and fac e s b eing equa l in volume) pr eserving uniformity . Delaunay tria ngulation bec omes Delaunay tessella tion in R d , provided that no more tha n d + 1 p oints be ing cospher ical (lying on the boundar y of the same sphere). In particular, with d = 3, the general s implex is a tetrahedron (4 vertices, 4 triangular faces and 6 edges ), which can b e mapped into a regular tetrahedro n (4 faces are equilatera l triangles) with v ertices (0 , 0 , 0) , (1 , 0 , 0 ) (1 / 2 , √ 3 / 2 , 0) , (1 / 2 , √ 3 / 6 , √ 6 / 3). L e t γ n ( r , d ) b e the domination nu mber for the extension to R d . Then it is easy to s ee that γ n ( r , 3) is nondegenerate as n → ∞ for r = 4 / 3, and otherwise degenerate. In R d , it c an b e seen that γ n ( r , d ) is nondegener ate in the limit only for r = ( d + 1) /d . Moreov er, it can b e shown that lim n →∞ P (2 ≤ γ n (( d + 1) /d, d ) ≤ d + 1) = 1, and we conjecture that lim n →∞ P ( d ≤ γ n (( d + 1) /d, d ) ≤ d + 1) = 1 . 7.1 Discussion In this article we inv estigate the mathematical prop erties o f a domination num b er metho d for the a nalysis of spatia l p oint patterns. The firs t proximit y map rela ted to r -factor proximit y ma p, N r Y , in liter ature is the spheric al pr oximity map , N S ( x ) := B ( x, r ( x )), (which is ca lled CCCD in the literature, see [Prieb e et a l., 20 01 ], [DeVinney et al., 2 0 02 ], [Marchette and Pr iebe, 200 3], [P riebe et al., 20 03a ], and [Pr iebe et a l., 2 003b]) . A s ligh t v a riation of N S is the ar c-s li c e pr ox imi ty map N AS ( x ) := B ( x, r ( x )) ∩ T ( x ) where T ( x ) is the Delaunay cell that c o n tains x (see [Ceyhan and P riebe, 20 03a ]). F ur ther more, Ceyha n and Pr iebe int ro duced the cen tral similarity proximit y map, N C S , in [Ceyha n and P riebe, 200 3a ]. The r -fa c to r proximit y map, when compared to the other s , has the a dv antages that the a symptotic distribution of the domination n umber γ n ( r ) is tractable (see Theorem 3). The distribution of the domination num b er of the proximit y catch digra phs based on N S or N AS is not tractable, a nd that o f N C S is an o p en problem. F ur ther more, N C S and N r Y enjoy the g eometry inv aria nce prop ert y over triangles for uniform data. Moreov er , while finding the exa ct minimum do minating sets is a n NP-Hard pro blem for N S , N AS , and N τ C S , the ex a ct minimum dominating sets can b e found in po lynomial time for N r Y . Additionally , N AS ( x ), N r Y ( x ), and N τ C S ( x ) are w ell defined only fo r x ∈ C H ( Y ), the co n vex hu ll o f Y , whereas N S ( x ) is well defined for a ll x ∈ R d . The N S (the proximit y map asso ciated with CCCD) is used in classifica tion in the litera ture, but not for testing s patial patterns b et ween two or more clas s es. W e develop a technique to test the patterns of segreg ation or asso ciation. There ar e many tests av ailable for segreg ation a nd a ssocia tion in ecolog y literature. See [Dixon, 1 994] for a sur vey on these tests and relev ant references . Tw o of the mos t commonly 10 used tests are Pielou’s χ 2 test of indep endence and Ripley’s test bas ed o n K ( t ) and L ( t ) functions. How ever, the test we intro duce here is no t compar able to either of them. Our metho d deals with a s ligh tly different t yp e of data than most metho ds to exa mine spatial patterns. The sample size for one type of p oint (t yp e X po in ts) is mu ch la rger compared to the the other (type Y p o in ts). The null hypo thesis we consider is considerably mo re restr ictiv e than current a pproaches, which can b e used m uch mor e generally . The null h y p othesis for testing seg regation o r asso ciation can b e desc r ibed in two slightly different fo r ms ([Dixon, 1 994]) : (i) co mplete spatial r andomness, that is, each c lass is distributed randomly throughout the ar e a o f interest. It des cribes b oth the arrang emen t of the lo cations a nd the asso ciation b et ween classes . (ii) ra ndom lab eling o f lo cations, which is less restrictive than s patial ra ndomness, in the sense that ar- rangement of the lo cations can either be random o r non-rando m. Our test is closer to the former in this r egard. References [Ceyhan and P riebe, 20 03a] Ceyhan, E . and P riebe, C. (2003a). Ce n tra l similarity proximit y ma ps in Delau- nay tessellations. In Pr o c e e dings of the Joint St atistic al Me eting, Statistic al Computing S e ction, Americ an Statistic al Asso ciation . [Ceyhan and P riebe, 20 03b] Ceyhan, E . and Prieb e, C. (20 03b). The us e of domination num b er of a ran- dom proximit y ca tc h digra ph for testing segreg ation/asso ciation. T echnical Rep ort 64 2, Depar tmen t o f Applied Ma thematics and Sta tis tics , J ohns Hopkins Universit y , Baltimore, MD 212 18-2682. s ubmitted for publication. [Co omes et al., 1999 ] Co omes, D. A., Rees, M., and L., T. (19 99). Identif ying a ggregation and a ssocia tion in fully mapp ed spatial data . Ec olo gy , 8 0:554–56 5 . [DeVinney et a l., 2002 ] DeVinney , J., Priebe, C. E ., Marchette, D. J., and So- colinsky , D. (2 002). Random walks and catc h digraphs in classification. http:/ /www.galaxy . gmu.edu/interface/I02/I2002Proceedings/DeVinneyJason/DeVinneyJason.paper.pdf . Computing Science a nd Statistics, V o l. 34 . [Dixon, 19 94] Dixon, P . M. (1994 ). T esting spa tial s e gregation using a neares t-neigh bo r contingency table. Ec olo gy , 75(7):194 0 –1948. [Gotelli and Graves, 1996 ] Gotelli, N. J. and Graves, G. R. (19 96). Nu l l Mo dels in Ec olo gy . Smithsonian Institution Pre s s. [Marchette and Pr iebe, 200 3 ] Marchette, D. J. and P riebe, C. E . (2003 ). Character izing the scale dimens io n of a high dimensio nal class ification problem. Pattern R e c o gnition , 36(1):45– 60. [Prieb e et al., 2001 ] Prieb e, C. E., DeVinney , J. G., a nd Ma rc hette, D. J. (200 1). On the distr ibution of the domination num b er of ra ndom class catch cover digraphs. Statistics and Pr ob ability L etters , 55 :239–246. [Prieb e et al., 2003 a] Prieb e, C. E., Marchette, D. J ., DeVinney , J., a nd So colinsky , D. (2003a ). Cla ssification using class cover catch digraphs. Journal of Classific ation , 20(1 ):3–23. [Prieb e et al., 2003 b] Prieb e, C. E., So lk a, J . L., Marchette, D. J., a nd Clark, B. T. (2003b). Class cov er catch digraphs for latent class discov ery in gene expr ession monitoring by DNA microar ra ys. Computational Statistics and Data Analysi s on Visualization , 43-4:6 21–632. [W est, 20 0 1] W est, D. B. (20 01). In tr o duction to Gr aph The ory, 2nd e d . Prentice Hall, NJ. 11 8 App endix Pro of of P roposition 1 T o prov e Pro position 1, w e show that the exp ected lo cus of the boundar y of the Γ 1 -region, ∂ (Γ r 1 ( X n )), go es to ∂ ( ς r Y ) as n → ∞ by showing that the exp ected loc i of X e j are e j for j = 1 , 2 , 3. See [Ceyhan and P riebe, 2003b] for the details . F or sufficiently lar ge n and g iv en X e j = ( x j , y j ) for j = 1 , 2 , 3, A (Γ 3 / 2 1 ( X n )) = √ 3 / 9(3 x 2 2 − 6 x 2 + 2 √ 3 y 2 x 2 − 2 √ 3 y 2 + y 2 2 + 3 + y 3 2 − 2 √ 3 y 3 x 3 + 3 x 3 2 + 4 y 1 2 ) . The asy mptotica lly accurate joint p df of X e j ’s is f 3 ( ζ ) = n ( n − 1)( n − 2)  √ 3 / 36( − 2 √ 3 y 1 + √ 3 y 3 − 3 x 3 + √ 3 y 2 + 3 x 2 ) 2  n − 3 / ( √ 3 / 4) n with the supp ort D S = { ζ = ( x 1 , y 1 , x 2 , y 2 , x 3 , y 3 ) ∈ R 6 : ( x j , y j )’s are distinct } . T he n for sufficiently large n , E [ A (Γ 3 / 2 1 ( X n ))] ≈ R D S A (Γ 3 / 2 1 ( X n )) f 3 ( ζ ) dζ , w hich g o es to 0 as n → ∞ at the rate O ( n − 2 ). See [Ceyhan and P riebe, 20 03b ] for the details. Pro of of Theorem 3 W e know that γ n ( r ) ≤ 3 a.s. for all r ∈ [1 , ∞ ] and all n . First we show that lim n →∞ P ( γ n (3 / 2) > 1) = 1. Note that P ( γ n (3 / 2) > 1) = P ( X n ∩ Γ 3 / 2 1 ( X n ) = ∅ ). Then w e find P ( X n ∩ Γ 3 / 2 1 ( X n ) = ∅ , E 2 ( n, ε )) where E 2 ( n, ε ) is the event such that 2 ε √ 3 ≤ X 1 ≤ 1 − 2 ε √ 3 and 0 ≤ Z 1 ≤ ε , and 1 / 2 ≤ X 2 ≤ 1 − 2 ε √ 3 , √ 3(1 − X 2 ) − ε ≤ Z 2 ≤ √ 3(1 − X 2 ), and 2 ε √ 3 ≤ X 3 ≤ 1 / 2 , and √ 3 X 3 − ε ≤ Z 3 ≤ √ 3 X 3 . First letting n → ∞ , then ε → 0, yields the desired res ult. See [Cey han and Pr iebe, 2003 b ] for the details . Next, lim n →∞ P ( γ n (3 / 2) ≤ 2) = lim n →∞ P ( γ n (3 / 2) = 2), since lim n →∞ P ( γ n (3 / 2) = 1) = 0. Let Q j := arg min x ∈X n ∩ R ( y j ) d ( x, e j ) = argmax x ∈X n ∩ R ( y j ) d ( ℓ ( y j , x ) , e j ) where e j is the edge opp osite vertex y j for j = 1 , 2 , 3 and let q j = ( x j , y j ) be the r ealization of Q j for j = 1 , 2 , 3 . Then γ n (3 / 2) ≤ 2 iff X n ⊂ N 3 / 2 Y ( Q 1 ) ∪ N 3 / 2 Y ( Q 2 ) or X n ⊂ N 3 / 2 Y ( Q 1 ) ∪ N 3 / 2 Y ( Q 3 ) or X n ⊂ N 3 / 2 Y ( Q 2 ) ∪ N 3 / 2 Y ( Q 3 ). Let the even ts E i,j := X n ⊂ N 3 / 2 Y ( Q i ) ∪ N 3 / 2 Y ( Q j ) for ( i, j ) = { (1 , 2) , (1 , 3) , (2 , 3 ) } . Then P ( γ n (3 / 2) ≤ 2) = P ( E 1 , 2 )+ P ( E 1 , 3 )+ P ( E 2 , 3 ) − P ( E 1 , 2 ∩ E 1 , 3 ) − P ( E 1 , 2 ∩ E 2 , 3 ) − P ( E 1 , 3 ∩ E 2 , 3 )+ P ( E 1 , 2 ∩ E 1 , 3 ∩ E 2 , 3 ) . By symmetry , P ( E 1 , 2 ) = P ( E 1 , 3 ) = P ( E 2 , 3 ) and P ( E 1 , 2 ∩ E 1 , 3 ) = P ( E 1 , 2 ∩ E 2 , 3 ) = P ( E 1 , 3 ∩ E 2 , 3 ). Hence P ( γ n (3 / 2) ≤ 2) = 3 h P ( E 1 , 2 ) − P ( E 1 , 2 ∩ E 1 , 3 ) i + P ( E 1 , 2 ∩ E 1 , 3 ∩ E 2 , 3 ) . W e find P ( E 1 , 2 ), by finding the a symptotically a ccurate joint p df of Q 1 , Q 2 . Let T ( Q j ) b e the triang le formed by the median lines a t y k and y l for k , l 6 = j and ℓ ( y j , Q j ), and let ε > 0 b e small enough s uc h that T ( Q j ) ⊂ R ( y j ), for j = 1 , 2 , 3. Then the asymptotically accura te joint p df of Q 1 , Q 2 is f 1 , 2 ( x 1 , y 1 , x 2 , y 2 ) = n ( n − 1) 1 A ( T ( Y )) 2  A ( T ( Y )) − A ( T ( q 1 )) − A ( T ( q 2 )) A ( T ( Y ))  n − 2 where A ( T ( q 1 )) = √ 3 / 36  − 2 √ 3 + 3 y 1 + 3 √ 3 x 1  2 and A ( T ( q 2 )) = √ 3 / 36  − 3 y 2 − √ 3 + 3 √ 3 x 2  2 with domain D I = { ( x 1 , y 1 ) ∈ R ( y 1 ) : y 1 ≥ − √ 3 3 + √ 3 x 1 + √ 3 ε, ( x 2 , y 2 ) ∈ R ( y 2 ) : y 2 ≤ − √ 3 3 + √ 3 x 2 − √ 3 ε } with ε > 0 b e small enoug h s uc h that T ( Q j ) ⊂ R ( y j ), for j = 1 , 2 , 3. 12 Then P ( E 1 , 2 ) ≈ . 4126 (which is found num erically ). See [Ceyhan and Pr iebe, 200 3b ] for the deta ils. Similarly w e find P ( E 1 , 2 ∩ E 1 , 3 ), by finding the joint p df of Q 1 , Q 2 , Q 3 , wher e T ( q 3 ) is the triangle with vertices 1 3 ( √ 3 − 3 y 3 ) √ 3 , y 3 ) , (1 / 2 , √ 3 / 6) , ( √ 3 y 3 , y 3 ). Then the asymptotically accura te joint p df o f Q 1 , Q 2 , Q 3 is f 123 ( x 1 , y 1 , x 2 , y 2 , x 3 , y 3 ) = n ( n − 1) ( n − 2) 1 A ( T ( Y )) 3  A ( T ( Y )) − A ( T ( q 1 )) − A ( T ( q 2 )) − A ( T ( q 3 )) A ( T ( Y ))  n − 3 where A ( T ( q 3 )) = √ 3 36 ( − √ 3 + 6 y 3 ) 2 with domain D I = { ( x 1 , y 1 ) ∈ R ( y 1 ) : y 1 ≥ − √ 3 3 + √ 3 x 1 + √ 3 ε, ( x 2 , y 2 ) ∈ R ( y 2 ) : y 2 ≥ − √ 3 3 + √ 3 x 2 − √ 3 ε , ( x 3 , y 3 ) ∈ R ( y 3 ) : y 3 ≤ √ 3 6 + ε } . Then P ( E 1 , 2 ∩ E 1 , 3 ) ≈ . 2 009 (see [Ceyhan and Prie b e, 2003 b ] for the details.) Likewise, we find P ( E 1 , 2 ∩ E 1 , 3 ∩ E 2 , 3 ) ≈ . 1 062 (see [Ceyha n and P riebe, 20 0 3b ] for the details.) Hence we get lim n →∞ P ( γ ( X n , N 3 / 2 Y ) = 2) ≈ . 7413, a nd lim n →∞ P ( γ ( X n , N 3 / 2 Y ) = 3) ≈ . 2587. 13